9 problems

Consider an annulus of thickness $\tau$, and write $\delta=A/|\Omega|$ for the prescribed area fraction. Let $D=D_{\alpha,A}$ be an optimal configuration, and let symmetry breaking…

Let $\operatorname{Ann}$ be the annulus semigroup and let … be the stated central extension. Diffeological universality conjecture. This extension is a universal central extension…

Universality conjecture. This central extension is a universal central extension. The claim concerns universality in the category of central extensions of the annulus semigroup. Th…

Nitsche-threshold conjecture. One has $r_\diamond<R$, and for every $r>r_\diamond$ there exists a minimizing $\rho$-harmonic diffeomorphism from $\mathbb{A}(r,1)$ onto…

Let $\mathbb{A}(r,R)$ and $\mathbb{A}(r_\ast,R_\ast)$ be spherical rings in $\mathbf{R}^n$, and let $\mathcal{F}$ be the family of homeomorphisms between them belonging to…

Let $\mathbb{A}$ and $\mathbb{B}$ be the annular domains appearing in the class $\mathcal{F}(\mathbb{A},\mathbb{B})$ of admissible mappings, and let…

Let $A(1,R)$ and $A(1,R_\ast)$ be annuli, and let $\rho$ be the weight defining $(\rho,n)$-harmonicity. For an increasing radial diffeomorphism between them, write … and set…

Nitsche bound. Then

The regular-metric annulus conjecture. Under these hypotheses,